Real Analysis : Twice Differentiable Functions

Let g:[0,1]->R be twice-differentiable (i.e both g and g' are differentiable functions) with g''(x)>0 for all x belong to [0,1].if g(0)>0 and g(1)=1 show that g(d)=d for some point d belong to (0,1) if and only if g'(1)>1.

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Proof:
"=>" Suppose g(d)=d for some d in (0,1). Since g"(x)>0 for all x in [0,1], then g'(x) is increasing. According to the Mean Value Theorem, we can ...